Subspaces

Discuss whetehr or not:

(a) R2 is a subspace of R3 over R
(b) {(0,0,0)} is a subspace of R3 over Q
(c) Q3 is a subspace of R3 over R
(d) RxQx{0} is a subspace of R3 over Q

I understand the requirements of a subset; however, I do not understand what it means for a set to be over say R. What does this mean and how does it play into deciding if one set is a subspace of another?

I understand the requirements of a subset; however, I do not understand what it means for a set to be over say R. What does this mean and how does it play into deciding if one set is a subspace of another?

Let me begin with a definition.Definition:An abelian group is a "vector-space" over field . When there exists a binary operation: such as,.Definition:If subset of a vector space is a vector space over the same field. It is called a "subspace".
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Originally Posted by TexasGirl

(a) R2 is a subspace of R3 over R

No, because we need that be a subgroup of but that is not true because are pairs, while are triples. Thus, these sets have totally differenet elements.

Originally Posted by TexasGirl

(b) {(0,0,0)} is a subspace of R3 over Q

Yes, because the binary operation we defined on over stasfies conditions 2,3,4,5. It only is questionable whether or not this binary operation is closed for over , we can see that it is closed. And finally is your trivial subgroup of thus, is a subspace of over.

Originally Posted by TexaxGirl

(c) Q3 is a subspace of R3 over R

No, because what happens if is irrational. Then, the binary operation is not closed because rationals multiplied by irrationals become irrationals.

Originally Posted by TexasGirl

(d) RxQx{0} is a subspace of R3 over Q

Yes, again properties 2,3,4,5. Are true because they depend on the same binary operations that are on . Checking, whether it is closed. Notice when multiplied by a rational numbers becomes, notice that is still real and is still rational. Thus, is a subspace of over